Lower Bounds for the Capture Time: Linear, Quadratic, and B - PowerPoint Presentation

Slide1

Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond Slide2

How to catch a robber on a graph?

The game of Cops and RobbersSlide3

The rules of the gameSlide4

The

Cop is placed first

CSlide5

The

Robber may then choose a placement

C

RSlide6

Next, they

alternate in moves

C

RSlide7

Next, they

alternate in moves

C

RSlide8

Next, they

alternate in moves

C

RSlide9

Next, they

alternate in moves

C

RSlide10

The

Cop won!

CSlide11

Graphs

where

cop wins have a

cop number

of

The

C

op won!

C

 Slide12

For graphs with

:

nodes:

moves always suffice

graphs

where

moves are needed

(Gavenčiak, 2010)

 How many moves does the cop need?Slide13

moves

suffice

for

paths Catch multiple?

R

R

R

CSlide14

moves for

the first

robber

Every further robber:

Cop moves to start in at most diameter

D

moves moves for the next robber

moves

in total 

Upper bound to catch ℓ

robbersSlide15

Lower bound

to

catch

ℓ

robbers

…

…Slide16

Lower bound to catch

ℓ

robbers

C

…

…Slide17

Lower bound

to catch

ℓ

robbers

C

R

R

R

…

…Slide18

Lower bound to catch

ℓ

robbers

C

R

R

R

…

…Slide19

Lower bound

to catch

ℓ

robbers

C

R

R

R

…

…Slide20

Lower bound

to catch

ℓ

robbers

R

C

R

R

…

…Slide21

Lower bound

to catch

ℓ

robbers

R

C

R

R

…

…Slide22

Lower bound

to catch

ℓ

robbers

C

R

R

R

…

…Slide23

Lower bound

to catch

ℓ

robbers

C

R

R

R

…

…Slide24

Lower bound

to catch

ℓ

robbers

R

C

R

R

…

…Slide25

Lower bound to catch

ℓ

robbers

R

C

R

…

…Slide26

Lower bound to catch

ℓ

robbers

R

C

…

…Slide27

Lower bound to catch

ℓ

robbers

R

C

…

…Slide28

Lower bound

to catch

ℓ

robbers

C

R

R

…

…

moves

per

robber

 Slide29

cop and

robbers

(in

graphs)

moves always suffice

needed in some graphs

Summary so farSlide30

cops and

robber

(in

graphs)

Best known upper bound:

(Berarducci and Intrigila, 1993)

Lower bound? What about

multiple cops and one robber

?Slide31

Let’s start with

two

cops and

one

robber

…

 Slide32

Let’s start with

two

cops and

one

robber

R

C

C

…

one

cop not enoughSlide33

moves

are

needed

Let’s start with

two

cops and

one

robber

C

R

C

…

 Slide34

How

large can

be compared to

?

Beyond

two

cops?

…

?

?Slide35

Aigner

and Fromme 1984: 3 for planar graphsMeyniel’s conjecture

(1985):

Known upper bound:

(

Chiniforooshan

2008)

Improved to

(Frieze,

Krivelevich

, and

Loh

2012

;

Lu and Peng 2012; Scott and

Sudakov

2011)

Pralat

(2010):

Beyond

two

cops?Slide36

Robber chooses side with less than

cops

Construction has

nodes

moves

are

needed

Multiple

cops and

one robber

…

Pralat (2010)

Pralat

(2010)

Note that

may hold!

 Slide37

cop and

robbers

(in

graphs)

moves always suffice

needed in some graphs

cops

and

robber

(in

graphs)

Best known upper bound:

moves

with

nodes

Summary so farSlide38

cops

and

robbers

(in

graphs

)

?

What about multiple cops and multiple robbers?Slide39

Are we done?

Multiple

cops and

multiple

robbers

…

…

…

…

 Slide40

Multiple

cops and multiple robbers

…

…

…

…

Problem:

?

 Slide41

Multiple

cops and multiple robbers

…

…

…

…

Problem:

?

C

Prevents robbers

from crossing

Slide42

How to deal with cop

?

Multiple paths do not help much:

Cop

„

emulates

“

robbers

Catches fraction each crossing

C

…

…Slide43

How to deal with cop

?

Multiple paths do not help much:

Cop

„

emulates

“

robbers

Catches fraction each crossing

Better idea:

Use a ring

C

…

…Slide44

How to deal with cop

?

Multiple paths do not help much:

Cop

„

emulates

“

robbers

Catches fraction each crossing

Better idea:

Use a ring

CSlide45

How to deal with cop

?

Multiple paths do not help much:

Cop

„

emulates

“

robbers

Catches fraction each crossing

Better idea:

Use a ring

CSlide46

How to deal with cop

?

Multiple paths do not help much:

Cop

„

emulates

“

robbers

Catches fraction each crossing

Better idea:

Use a ring

CSlide47

Construction of the ring

…

…

…

…

…

 Slide48

Robber placement

Robbers

choose side with less cops

R

R

RSlide49

Robber strategy

cops needed to

catch

robber in gadget graph

If

, then all other robbers escape “down”

R

R

RSlide50

Robber

strategy

cops needed to

catch

robber in gadget graph

If

, then all other robbers escape “down”

But if

?

R

R

RSlide51

Robber

strategy

cops needed to

catch

robber in gadget graph

If

, then all other robbers escape “down”

But if

?

R

R

R

C

C

an catch half of the robbers!Slide52

Robber

strategy

R

R

R

CSlide53

Robber

strategy

R

R

R

CSlide54

Robber

strategy

R

R

R

CSlide55

Robber

strategy

R

R

R

C

Cops need

moves

to

catch

2

robbers

moves

to

catch

all

robbers

 Slide56

cop and

robbers

(in

graphs)

moves always suffice

needed in some graphs

cops

and

robber

(in

graphs)

Best known upper bound:

moves

with

nodes

cops

and

robbers

(in

graphs)

moves

with

More

than

robbers?

SummarySlide57

Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond